# Numerical Methods for Partial Differential Equations

## Final Projects

To download postscript files of the project reports, click here, which is also a list of abstracts for the project presentations given in class on May 08 and 10, 2001.

Original announcement of the project presentations: This class included a very substantive project that each student was supposed to work on for at least about six weeks. Over two days of classes (on May 08 and 10), the projects will be presented; see the program for project presentations for details. During that time, we will also perform a peer-review (among the students in the class) of the reports. The final, edited reports will be posted as postscript files and linked through the program for project presentations.

Homework scores ordered by the last four digits of your student number: Math 621

## Basic Information

• Matthias K. Gobbert, Math/Psyc 416, (410) 455-2404, gobbert@math.umbc.edu,
office hours: TTh 03:00-03:50 or by appointment
• Lectures: TTh 07:00-09:00, MP 401
• Prerequisite: Math 630, familiarity with a high-level programming language and basic knowledge of the UNIX operating system, corequisite: Math 620, or instructor approval
• Textbook: Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. A copy of the textbook is on reserve in the library.
 Homework Midterm Final 40% 30% 30%
The homework is weighted so heavily, because it includes the computer assignments that are vital to understanding the course material. See the general policies and procedures for more information.

## Overview

Realistic models for physical processes in nature and in engineering necessarily involve partial differential equations. The models are as varied as reality itself, but most often non-linear and often involving systems of partial differential equations. In all but some textbook examples, an analytic solution is impossible. That necessacitates the use of numerical methods for partial differential equations, and this area forms a vast field itself and is the driving force behind many other fields like numerical linear algebra, scientific computing, and the development of parallel computers.

However, despite their many forms, many equations share certain fundamental mathematical properties and can be classified into the three basic categories of elliptic, parabolic, and hyperbolic partial differential equations. It makes therefore sense to study the mathematical properties and numerical methods for prototype equations of each type.

This class will provide an overview of the types of equations, their most fundamental mathematical properties, and demonstrate numerical methods for them. The most basic methods are finite difference and finite element methods, and we will discuss both types for each prototype equation. We will use this as the basis for discussing the associated issues of discretizing the time-direction and solving large sparse systems of linear equations efficiently with respect to memory and computing time.

It is not possible to cover this subject in one semester. Hence, students should be prepared to do some independent research as part of the homework. The class will also entail a very substantive final project involving a much broader problem than a homework would, for instance, an engineering design problem, where the numerics are used to answer a question quantitatively. The final project will culminate in a written report and an oral presentation on the level of a capstone experience for a Master of Science degree without a thesis.

To allow sufficient time for work on the final project and to enable the attendance of part-time students, the lectures will be mostly concentrated in the first half of the semester, while lengthened to two hours; see the syllabus for details.